Bayesian Methodologies for Big Spatial Data That Avoids Covariance Matrix Inversion

Spatial datasets are becoming increasingly more common over the recent decades. Rapid devel- opments in technology has brought an abundance of information and data. Big spatial datasets produce many computational challenges. In this essay, we focus on modeling big Gaussian and non-Gaussian spatial datasets. It is increasingly understood that the assumption of stationarity is unrealistic for many spatial processes. In Chapter (2), we combine dimension expansion with a spectral method to model big non-stationary spatial fields in a computationally efficient manner. Specifically, we use Mej ́ıa and Rodr ́ıguez-Iturbe [1974]’s spectral simulation approach to simulate a spatial process with a covariogram at locations that have an expanded dimension. We introduce Bayesian hierarchical modelling to dimension expansion, which originally has only been modeled using a method of moments approach. In particular, our algorithm is a type of collapsed Gibbs sampler and it involves two steps. Our method is both full rank and non-stationary, and can be applied to big spatial data because it does not involve storing and inverting large covariance matrices. Additionally, we have fewer parameters than many other non-stationary spatial models. We demonstrate the wide applicability of our approach using a simulation study, and an application using ozone data obtained from the National Aeronautics and Space Administration (NASA). In Chapter (3), we propose a method for prediction using spatial count data that can be reasonably modeled assuming the Conway-Maxwell Poisson distribution. Modeling spatial count processes is often a challenging task. That is, in many real-world applications the complexity and high dimensionality of the data do not allow for routine model specifications. For example, spatial count data often display over-dispersion or under-dispersion. In order to allow for such structure, we propose a computationally efficient hierarchical Bayesian model. We develop the Conway Maxwell- Poisson model (COM-Poisson), which is a two parameter generalisation of the Poisson distribution. The COM-Poisson distribution allows for the flexibility needed to model count data that are either over or under-dispersed. The limiting factor of the COM-Poisson distribution is that the likelihood function contains multiple intractable normalizing constants and is not always feasible for MCMC techniques. We provide discussion on a project that addresses these issues for spatial count data. Finally, we provide some discussions and future works in Chapter (4).